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Grothendieck’s mathematics and Child Born of Water

Saturday, December 13th, 2014

[ by Charles Cameron — two approaches to mathematics, two types of heroism, and their respective complementarities ]
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I wish to propose a clear analogy between the mathematician Grothendieck‘s two styles of approach to a problem in mathematics, and the Navajo Twin Gods, Monster-Slayer and Child-Born-of-Water.

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Twins

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Steve Landsburg‘s post, The Generalist, compares two approaches to mathematics, as practiced by two eminent mathematicians:

If there was a nut to be opened, Grothendieck suggested, Serre would find just the right spot to insert a chisel, he’d strike hard and deftly, and if necessary, he’d repeat the process until the nut cracked open. Grothendieck, by contrast, preferred to immerse the nut in the ocean and let time pass. “The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough.”

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In the paras leading up to this one, Landsburg gives us the insight that these two approaches can be generalized as “zooming in” and “zooming out”:

Imagine a clockmaker, who somehow has been oblivious all his life to many of the simple rules of physics. One day he accidentally drops a clock, which, to his surprise, falls to the ground. Curious, he tries it again—this time on purpose. He drops another clock. It falls to the ground. And another.

Well, this is a wondrous thing indeed. What is it about clocks, he wonders, that makes them fall to the ground? He had thought he’d understood quite a bit about the workings of clocks, but apparently he doesn’t understand them quite as well as he thought he did, because he’s quite unable to explain this whole falling thing. So he plunges himself into a deeper study of the minutiae of gears, springs and winding mechanisms, looking for the key feature that causes clocks to fall.

It should go without saying that our clockmaker is on the wrong track. A better strategy, for this problem anyway, would be to forget all about the inner workings of clocks and ask “What else falls when you drop it?”. A little observation will then reveal that the answer is “pretty much everything”, or better yet “everything that’s heavier than air”. Armed with this knowledge, our clockmaker is poised to discover something about the laws of gravity.

Now imagine a mathematician who stumbles on the curious fact that if you double a prime number and then halve the result, you get back the number you started with. It works for the prime number 2, for 3, for 5, for 7, for 11…. . What is it about primes, the mathematician wonders, that yields this pattern? He begins delving deeper into the properties of prime numbers…

Like our clockmaker, the mathematician is zooming in when he should be zooming out. The right question is not “Why do primes behave this way?” but “What other numbers behave this way?”. Once you notice that the answer is all numbers, you’ve got a good chance of figuring out why they behave this way. As long as you’re focused on the red herring of primeness, you’ve got no chance.

Now, not all problems are like that. Some problems benefit from zooming in, others from zooming out. Grothendieck was the messiah of zooming out — zooming out farther and faster and grander than anyone else would have dared to, always and everywhere. And by luck or by shrewdness, the problems he threw himself into were, time after time, precisely the problems where the zooming-out strategy, pursued apparently past the point of ridiculousness, led to spectacular, unprecedented, indescribable success. As a result, mathematicians today routinely zoom out farther and faster than anyone prior to Grothendieck would have deemed sensible. And sometimes it pays off big.

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I no longer have — alas — a copy of Where the Two Came to their Father, the first volume in the Bollingen Series, with its suite of 18 sand paintings beautifully rendered in silkscreen by Maud Oakes, but their respective black and blue colorations lead me to suppose that the illustration at the head of this post, taken rom that series, shows the twin heroes, Monster Slayer (black) and Child Born of Water (blue) whose journeys and initiation are the subject of the rituasl “sing” recorded in that book.

The theme of two male hero twins is central to the mythologies of the American continent, according to Jospeh Campbell, who contributed a commentary to Oakes’ recording of Jeff King‘s performance of this ceremony, and lacking both the King > Oakes > Campbell book and Gladys Reichard‘s two volumes on Navaho Religion, I must draw on brief quotes from miscellaneous web sources to dramatize the differences between the twins.

Monster Slayer is the doer of deeds, similar in nature to other masculine, not to say macho, heroes — while Child Born of Water is the contemplative of the pair:

The Sun [Jóhonaa’éí] gave them prayersticks and then told them that the younger of the two (Born for Water) would sit watching these prayersticks while the older (Monster Slayer) went out to kill the monsters. If these prayersticks began to burn, this would signal that his brother was in danger and that he should go to him to help.

Reichard explains:

Monster Slayer (na’ye’ ne’zyani) (I) represents impulsive aggression, whereas Child-of-the-water represents reserve, caution, and thoughtful preparation.

A measure of their respective strategies, and of the ways in which the insights of Child Born of Water can succeed where the brute force tactics of Monster SLayer fail, can be gleaned from this section of their story, also I believe taken from Reichard:

When The Twins visited Sun the second time, he said he was willing to help them, but this time he wanted them to return the favor: “I wish you to send your mother to the west that she may make a new home for me.” Whereupon Monster Slayer, believing himself equal to any task, replied, “I will do so.I will send her there.” Then Child-of-the-water reminded them both: “No, Changing Woman is subject to no one? we cannot make promises for her. She must speak for herself? she is her own mistress. But I shall tell her your wishes and plead for you.”

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One commentator glibly suggests that the joint presentation of the hero as twins is “a clever reminder that progress depends upon cooperation between our mind and our heart” — but the psychologist Dr Howard Teich offers a far more depthful interpretation: that the two twins represent two forms of masculine heroism, one the familiar macho hero of war movies, and the other wiser and subtler, the possessor of traits commonly attributed to the feminine — and hugely undervalued — in our culture.

Dr Teich suggests we must (urgently) abandon the division of virtues into “male” and “female” types, reognize that these types are complementary rather than rivalrous, that both are necessary functions of both males’ and females’ psyches, and begin to integrate the wholeness that both strategies together represent, in our own approaches to our lives in general, to the natural world around us, and indeed to warfare — unsurprisingly, since we first encounter the twins in the ceremonial specifically devised by the Navajo to protect young warriors on their way to battle, and to reintegrate them in harmony and balance on their return.

As Teich puts it:

Monster Slayer and Child Born of Water, as these Twin Heroes are called, are the most sacred of all the legendary heroes in Navaho mythology. It is rare for the Navaho even to speak of the twins; their presence is to be felt rather than observed, and their lessons absorbed rather than applied. Although the lessons the twins hold may be countless, their particular manifestation of a deeper, more complex image of masculinity deserves the reader’s especial attention.

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I’d like to suggest that in the same way that there are “zooming in” and “zooming out” styles in mathematics, and “monster-slayer” and “born of water” styles of heroism, there are in fact twin traditions of understanding the world which we might term scientific and poetic, or in Teich’s terms — and those of the alchemists — solar and lunar.

A unified or “solunary” vision will encompass the virtues of both.

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Dr Teich’s review of the King > Oakes > Campbell book under the title A Dual Masculinity was irst piublished in The San Francisco Jung Institute Library Journal, Vol. 13, No. 4, 1995. He now has a book out treating these themes: Solar Light, Lunar Light.

Oh, and please don’t expect me to know anything more about Grothendieck’s mathematics than I read in Landsburg’s article.

A DoubleQuote in the (Arctic) Wild

Friday, January 3rd, 2014

[ by Charles Cameron — always on the lookout for intriguing double-images ]
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There’s an implied “this is to that as this is to that” double analogy here. Just how well or ill it teaches coordinate systems I leave to others to decide — even without the analogical joking though, it’s an intriguing visual juxtaposition.

Gaming the Connections: from Sherlock H to Nada B

Sunday, December 29th, 2013

[ by Charles Cameron — the game of Connect the Dots in play and practice ]
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CIA's (now ret'd) Nada Bakos examines the Al Qaida board in the HBO docu, Manhunt

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Manhunt, the HBO documentary, does what (not having been there and seen that at the time) appears to be a decent job of recreating some of the cognitive stratregies employed by CIA officers in the OBL hunt. The one I’m interested in here is the building of a “link chart” or cognitive map — law enforcement “evidence board” — the idea being (a) to note known connections visibly, and (b) to encourage the mind to make intuitive leaps that reveal previously unknown connections between nodes… or “dots”.

Sophisticated software does this sort of thing algorithmically with regard to (eg) network connections via phone-calls, but the human mind is still better than AI at some forms of pattern recognition, and that’s the aspect that interests me here.

Aside:

For more on the cognitive significance of the link chart in Manhunt, see my post Jeff Jonas, Nada Bakos, Cindy Storer and Puzzles.

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Benedict Cumberbatch‘s Sherlock lays out the way it works —

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Okay, so one way to visualize connections is to make a fairly random collage of relevant photos, names, dates and places, and tie it together with links of string or ribbon. That’s the equivalent of what in HipBone games terms we’d call a “free-form” game, and it works well for the “divergent”, initial brainstorming phase of thought. But it does little to bottle its own energy, to focus down, to force the mind — in the no less powerful “convergent” phase — into perceiving even more links than occur spontaneously in building the link chart in question.

HipBone‘s preformatted boards take the cognitive process to that second stage. They work on one of the most powerful ingredients in creativity: constraint. Business writer Dave Gray of Communication Nation puts it like this:

Creativity is driven by constraints. When we have limited resources — even when the limits are artificial — creative thinking is enhanced. That’s because the fewer resources you have, the more you are forced to rely on your ingenuity.

But that premise doesn’t just hold true for business problem-solving — it’s at the heart of creative thinking at the Nobel level, too, in both arts and sciences. Consider mathematician Stanley Ulam, writing in his Adventures of a Mathematician:

When I was a boy I felt that the role of rhyme in poetry was to compel one to find the unobvious because of the necessity of finding a word which rhymes. This forces novel associations and almost guarantees deviations from routine chains or trains of thought. It becomes paradoxically a sort of automatic mechanism of originality…

Here’s how the poet TS Eliot puts it:

When forced to work within a strict framework the imagination is taxed to its utmost – and will produce its richest ideas.

A Hipbone Gameboard such as the Waterbird, Dartboard, or Said Symphony board is chosen precisely to challenge the mind with third, fourth and fifth rounds of “creative leaps” — thus adding both divergent and convergent cognitive styles to this form of graphical analysis.

That’s my point here — and a plug for HipBone-Sembl style thinking.

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I can’t resist adding a couple of instances in which the meme of “connecting the dots” via a link chart or evidence board has crept from TV series that I enjoyed into the world of games — this first one based on the terrific French detective series, Engrenages, retitled Spirals for British consumption:

— and this one for fans of the US TV series, Breaking Bad:

Leap Worlds: my 3QD attempt

Wednesday, December 11th, 2013

[ by Charles Cameron — another runup to the glass bead game ]
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Below you can read my submission to the wonderful 3 Quarks Daily “web aggregator” site as a candidate to join their regular Monday blogging team — it didn’t even make their “close but no cigar” list, but I wrote it and I like it, so I’m posting it here.
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They are both fine American authors, stylists of high repute yet little known, Annie Dillard and Haniel Long, but I hadn’t associated the two of them particularly closely until that day. I’d snarfed up a copy of Long’s Letter to Saint Augustine from the dollar box at Pasadena’s magnificent Archives theological bookstore, and on my way home to nearby Eagle Rock, stumbled on a passage that seemed hazily familiar.

My friend Jens Jensen, who is an ornithologist, tells me that when he was a boy in Denmark he caught a big carp embedded in which, across the spinal vertebrae, were the talons of an osprey. Apparently years before, the fish hawk had dived for its prey, but had misjudged its size. The carp was too heavy for it to lift up out of the water, and so after a struggle the bird of prey was pulled under and drowned. The fish then lived as best it could with the great bird clamped to it, till time disintegrated the carcass, and freed it, all but the bony structure of the talon.

By the time I arrived back at my books, I knew it was Annie Dillard I’d been reminded of, and a quick, no, excited but fumbling search turned up this passage from her Teaching a Stone to Talk:

And once, says Ernest Seton Thompson–once, a man shot an eagle out of the sky. He examined the eagle and found the dry skull of a weasel fixed by the jaws to his throat. The supposition is that the eagle had pounced on the weasel and the weasel swiveled and bit as instinct taught him, tooth to neck, and nearly won. I would like to have seen that eagle from the air a few weeks or months before he was shot: was the whole weasel still attached to his feathered throat, a fur pendant? Or did the eagle eat what he could reach, gutting the living weasel with his talons before his breast, bending his beak, cleaning the beautiful airborne bones?

They’re emblematic, those two quotes, the way I see them: emblematic in the sense that each could be the basis for a heraldic shield, the eagle stooping to carry off the weasel, air creature triumphant over creature of earth in the one, the fish dragging the bird down in the other — creature of water gathering the air creature into its own fatal realm.

Emblematic too, I see them, of the possibility of a rhyme between thoughts — and I have quoted them as such in previous essays, alongside rhymes of sound and rhymes of meaning, womb and tomb being the classic examples of choice, but also rhymes of image, the fan rotors and helicopter rotors in the opening sequences of Coppola‘s Apocalypse Now, rhymes of melody in counterpoint, in fugue… and rhymes, yes, in history, as when Sir Frederick Stanley Maude told the people of Baghdad in 1917, “Our armies do not come into your cities and lands as conquerors or enemies, but as liberators…” and Donald Rumsfeld chimed in, telling his own troops in that same weary city, almost a century later, “Unlike many armies in the world, you came not to conquer, not to occupy, but to liberate and the Iraqi people know this.”

Oh yeah?

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The world is woven, Jung tells us in one of his most original and illuminating insights, of woof and warp, causaland acausal principles , it is at once at every point synchronic and diachronic.

Moving through time, we have the causal, diachronic principle, one things leads to another: let us sort and analyze the actions of one thing on another until psychology becomes sociology, sociology turns into evolutionary biology, biology into genetics, genetics a form of chemistry that is essentially physics — and physics, at the quark-level at last, a matter of statistics, mathematics unfettered even by the dualism of wave and particle.

Let me be clear about this. I do not begrudge Higgs his boson or his Nobel, nor Chandrasekhar his Nobel or his limit. But both men followed the warp, the causal, time-bound length-wise threads of discovery to their fraying edges. And the causal threading of events, time’s warp in our lived universe, is the mode best suited to quantity, to the determinable, and knows little of mystery unless magnitude alone — the infinities of astronomy, the infinitessimals of subatomics, alone will qualify.

Oh, scale is a marvel, true enough — but it is quality, not quantity, where the mystery and the greater meaning resides.

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And so we come to the mind’s other faculty, the other manner in which the world is woven, the manner of rhyme and repetition, synchronicities and semblances, of patterns recognized within and across disciplines. The cross-weave.

This has been the step-child of cognition for too long — but with the rise of cybernetics, feedback loops, complexity theory, network thinking and multi-causality, we can no longer think only in linear progressions, but must also cultivate associative, lateral, sideways thinking — in short, creative leaps.

Creative leaps occur when we recognize commonalities across conceptual distances — theme and variations, as musicians would say, rhymes in the nature of things, multiple perspectives and voices in counterpoint. So the nature of our current world, in all its complexity and variegation, calls for what I would call a music of ideas.

I’m not the first to have this idea, Glenn Gould was pursuing it in his work for radio, blending the many voices and conversations in a train compartment, or at the different tables in a truck-stop café, to form an interwoven whole that comprehended all of its voices, all of its parts in a greater music. But it is Edward Said — another musicians, when he was not occupied with Israeli-Palestinian politics or literature — who observed in an essay in Power, Politics and Culture, p. 447:

When you think about it, when you think about Jew and Palestinian not separately, but as part of a symphony, there is something magnificently imposing about it. A very rich, also very tragic, also in many ways desperate history of extremes — opposites in the Hegelian sense — that is yet to receive its due. So what you are faced with is a kind of sublime grandeur of a series of tragedies, of losses, of sacrifices, of pain that would take the brain of a Bach to figure out. It would require the imagination of someone like Edmund Burke to fathom.

So that’s the symphonic scope of the thing, seeing the whole with all its fractures and dissonances as a music — a music that calls for harmonization, but remains in complex counterpoint.

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Hermann Hesse was the first great proponent of the music of ideas, at least in modern western times, and his glass bead game — evoked but never defined in his Nobel-winning novel of that name — offers us a glimpse both of the nature of moves and of the possible grandeur of an implicit world-architecture, formed of resonances and semblances, rather than of causes and effects.

Hesse gives us an insight into how the game is played at the level of moves and themes when he says a given game might have explored “the rhythmic structure of Julius Caesar’s Latin and discovered the most striking congruences with the results of well-known studies of the intervals in Byzantine hymns” — but he could hardly have known, back in the 1940s, that in 1978 the University of Wisconsin Press would publish Jane-Marie Luecke OSB’s monograph, Measuring Old English Rhythm: an Application of the Principles of Gregorian Chant Rhythm to the Meter of Beowulf. Intentionally or unintentionally, his game is played by all whose minds play with meanings. And there’s a game move right there, in the conviviality between the fictional move of Hesse himself, and the monograph, years later, of a Benedictine nun.

Tiny, you say, a tiny move — but a move in what Hesse termed the “hundred-gated Cathedral of Mind.”

And the scope of that cathedral, in its gradual entirety, is vast — encompassing all the vast quantities of the sciences, with the qualitative depths and heights of the arts and humanities too:

The Glass Bead Game is thus a mode of playing with the total contents and values of our culture; it plays with them as, say, in the great age of the arts a painter might have played with the colors on his palette. All the insights, noble thoughts, and works of art that the human race has produced in its creative eras, all that subsequent periods of scholarly study have reduced to concepts and converted into intellectual values the Glass Bead Game player plays like the organist on an organ. And this organ has attained an almost unimaginable perfection; its manuals and pedals range over the entire intellectual cosmos; its stops are almost beyond number. Theoretically this instrument is capable of reproducing in the Game the entire intellectual content of the universe.

We are drawing close, in a humanly possible way and without making any predications of the being or non-being of such a supposed entity, to the very mind of god.

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And with immediate real world application, as when Maxwell sees the commonality between electricity and magnetism, Kekulé the serpent-biting-it’s-tail like form of the benzene ring — or Taniyama‘s 1955 “surmise” as Barry Mazur puts it, that “every elliptic equation is associated with a modular form” — an insight that was to bear rich fruit forty years later, in Andrew Wiles‘ proof of Fermat’s Last Theorem.

Hesse’s vision of the game was lucid, elegant, intellectual — but lacked real world application. Viewed as a method of scoring the music of ideas, it offers illuminations from the most abstract and theoretical of mathematics to the most complex of opposed political intrangencies — from Tamiyama to Edward Said, and from Christopher Alexander‘s Pattern Language to John Holland‘s “genetic algorithms“. Alexander and Holland each indicate their debt to Hesse’s fictional game in their own respective works.

How, then, to notate this game, these moves that leap side-wise, pattern-wise, semblance-wise across boundaries and boxes, limitations and disciplines — for it is these leaps, as Arthur Koestler notes in his The Act of Creation, that give us the surprised aha! of discovery, the unexpected ha! of laughter, the gut-wrenching wail aiiyeee! when tragedy strikes.

My own inclinations favor play — solo or with a friend — at a coffee house, with pencil and paper napkin, with a small graph for a board, ideas played at its nodes, connections and resonances represented by its edges. I’m thinking of network mapping on a human scale, between seven and a dozen nodes, each one rich in meaning, anecdote, quote, statistic, image, snatch of song…

But the essence is the single move, the single resemblance. And for this I have a form which I would like to offer you, downloadable, for your own experiments:

Download the image and fill the two spaces with what you will — whatever you can sketch, whistle, count out or scribble —

I think you’ll find the greatest reward comes when the two ideas, visuals, verbals, aurals you juxtapose are closely related yet drawn from distantly separated regions of thought.

At their best, such juxtapositions cross galaxies. My own most cherished example to date compares a night sky by Van Gogh with a von Kármán diagram of turbulent flow…

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And where does this lead us? What becomes of CP Snow‘s famed Two Cultures?

Two great rows of pillars in Hesse’s hundred-gated cathedral, perhaps — best appreciated when one looks up, and sees the great vaulted roof, the magnificence of the arches between them.

In future Monday columns here on 3QD, I hope to bring you some further moves in the great game of correspondences, weaving together topics that have caught my attention in the preceding month — now in poetry, now on the morning news.

The leaps.

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I’m happy to report that friend Bill Benzon made the 3 Quarks Daily cut and will be blogging there, and that friend Omar Ali is already one of their regulars.

Of nested and coiled serpents in logic

Saturday, August 31st, 2013

[ by Charles Cameron — another exploration into forms of insight — in this case the Matrioshka effect, spiral staircases and the like, with a glance at holy winds and human fingertips ]
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A mention of blogs about blogs about blogs seems to me to qualify for the “nested serpent” category of forms that are worth watching out for, the nest (or spiral, from which I am guessing the nest is not entirely separable) being of particular interest because while seemingly simple enough, it all too often reaches at one end or both into the infinities, where paradox meets epiphany… as my second example will show.

But first, by sheer good fortune, I came across this verse from the book of Ecclesiastes as I was polishing this post for publication:

The winde goeth toward the South, and turneth about vnto the North; it whirleth about continually, and the winde returneth againe according to his circuits.

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That’s the pattern we’re looking for, and I ran across it recently in a comment my friend Allen Stairs made, and the response he received:

Okay, I didn’t follow — so I asked Allen for an explanation, and he wrote me:

Actually. “equipollent” was a bad choice of terms. “Equinumerous” wold have been better.

But the thing about numbers and those dolls: natural numbers have their identity intrinsically, so to speak. In set theory, one way to represent them is as the series

1 = {Ø}, 2 = {Ø,{Ø}}, 3 = {Ø,{Ø,{Ø}}}, etc.

In fact, we can even use the simpler construction

1 = {Ø}, 2 = {{Ø}}, 3 = {{{Ø}}}, etc.

So if we’re given the set, its structure tells us which number it is. I

Now a finite set of Russian dolls does much the same thing. We could count the innermost one as 1, the next as 2, the next as 3, and so on, and if you were given the doll, you’d be able to tell which number it represented. Or if we wanted, we could let the outermost doll represent 1, and work our way in. But if we take the set of all natural numbers, things get a little wonkier. The thing about a set of dolls is that there’s an outer one; the charm is in the fact that there’s a place to start opening them. So suppose we have an infinitely nested set of dolls. What number does the outermost one correspond to? It can’t be a natural number, because for any natural number, the nesting would have to be finite. It can’t be the infinite number Aleph-null because among other things, if the nesting is infinite downward then each doll has the same structure as the one that encloses it, and so it seems that there’s no way for the individual dolls to represent distinct integers.

Now if we’re given the whole set of dolls, there’s a sort of substitute: match dolls to numbers depending on how many “predecessors” they have. The outermost doll has no predecessors, so let it be 1; the next one in has 1 predecessor, so let it be 2. And so on. But we still have a problem: there’s nothing about the doll itself that tells us which integer it represents.

So my little point was that Harold’s joke was about “how many?” but the thing about the dolls is that they might seem at first to have the right structure to represent the natural numbers, and yet they don’t — at least, not the whole set of natural numbers.

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Did Ecclesiastes mention the winds? Here’s a discussion of wind spirals from David Avram‘s The Spell of the Sensuous: Perception and Language in a More-Than-Human World:

Although invisible, the Holy Wind can be recognized by the swirling and spiraling traces that it continually leaves in the visible world. The Winds that enter a human being leave their trace, according to the Navajo, in the vortices or swirling patterns to be seen on our fingertips and the tips of our toes, and in the spiraling pattern made by the hairs as they emerge from our heads. As one elder explains:

There are whorls here at the tips of our fingers. Winds stick out here. It is the same way on the toes of our feet, and Winds exist on us here where soft spots are, where there are spirals. At the tops of our heads some children have two spirals, some have only one, you see. I am saying that those (who have two) live by means of two Winds. These (Winds sticking out of the) whorls at the tips of our toes hold us to the Earth. Those at our fingertips hold us to the Sky. Because of these, we do not fall when we move about.

That last italicized quote is from James Kale McNeley, Holy Wind in Navajo Philosophy — a remarkable book for anyone interested in the holiness of spirit…


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